The 17 equations that changed the world

I was so excited the first time I read Ian Stewart’s book entitled “The 17 equations that changed the world“. The book is written in simple and easy to understand language with interesting practical examples for applications. I immediately wrote to Ian Stewart requesting if I could reproduce his work in form of posters in both English and French to be used at AIMS-IMAGINARY in Senegal in 2015 (see here). I remember his only reply was “please proceed but I won’t be able to attend since I have prior committments”. Business Insider published a list of these equations emphasing further how intuitive they are (see here). I do strongly believe every school in the world be it elementary, college, secondary, technical, university, you name it, should have these posted up or painted on the walls of their science departments/offices, classrooms, laboratories etc; in all langauges applicable.…

By Michael Kateregga| July 29th, 2016|Uncategorized|3 Comments

Square roots: in your head

I came across the following YouTube video a while back which uses a strange trick to accurately approximate square roots. I suggest watching at least the first minute where the presenter explains how it’s done:

Let’s do an example. Approximate to 2 decimal places: $\sqrt{40}$

First, YouTube tells us to find the nearest perfect square that’s less than 40, that’s 36, and take the root, giving us 6. So our answer is, obviously, 6 point something. That something is a fraction, where the numerator is the difference between 40 and 36, and the denominator is 2 times 6. So we have:

$\sqrt{40} \approx \sqrt{36} +\frac{4}{12} = 6+ \frac{1}{3} \approx 6.33$

So what’s the actual answer to 2 decimals? It’s 6.32. That’s what I like to call: pretty darn close. Naturally, there are a few catches to this technique: you need to know your perfect squares, you need to know your fractions, and things tend to get hard with larger numbers.

But still, I thought this was surprisingly effective for such a simple piece of arcane trickery.…

By Jean-Jacq du Plessis| June 23rd, 2016|Uncategorized|1 Comment

Type Theory, Logic, and Programming

Hello Internet. In this blog post we are going to learn some type theory, which is a field that lies in the intersection of mathematics and computer science. We are eventually going to learn the variant called Martin-Loef Type Theory, or MLTT for short. To motivate it, we are going to try answering the question, “What is a valid program?”

Not every program makes sense, for example what would a program consisting of the single expression x + 2 mean? If we don’t know what x is, then we don’t know what the meaning of that expression is. More importantly, a program can combine expressions that are incompatible with each other. For example, if + is the usual operation that adds two numbers then an expression like "abc" + 5 can’t be given a meaning. To avoid such meaningless programs, we should talk about the different kinds of data and the valid operations on these pieces of data.…

By Kabelo Moiloa| June 19th, 2016|Uncategorized|2 Comments

Down the rabbit hole

The following has been a rather interesting journey – from a test question which seemed fine, to a subtlety which seemed easy, to a discussion with a number of different mathematicians about the nature of distributions, measure theory and regularisation. I will try and make it as clear as possible in the post below. Note, as mentioned in the comments, we have actually only found the solution to this problem for a constrained range of x, and not x $\in \mathbb{R}$ . I didn’t want to complicate things any more than necessary here for first year students, but the comments are very important too.

In a recent class test, there was a question, written by me, which was not quite the question that I wanted to ask. It turns out that it does have an answer, but it’s not an answer that can yet be found by the means at the class’s disposal.…

By Jonathan Shock| May 27th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|7 Comments

On Convergent Sequences and Prime Numbers

Ever since Euclid first proved that there are infinitely many prime numbers, mathematicians have found ever more creative ways to prove the same result, and also various stronger theorems that imply it. Dirichlet’s Theorem, for example, states that if $m$ and $n$ are relatively prime integers, then there are infinitely many prime numbers of the form $mk + n$ for some integer $k$ . It is also known that the sum of the reciprocals of the prime numbers diverges, that the sum

$\displaystyle \sum_{\substack{p \leq n \\ p \text{ prime}}} \frac{1}{p} \sim \log(\log(n))$

and that the number of prime numbers less than $n$ is asymptotically equal to $\displaystyle \frac{n}{\log(n)}$ . In this blog post, we will continue this proud tradition by proving that there are infinitely many prime numbers which have your phone number somewhere in their digits, and which simultaneously have a prime number of digits.

To do so, we will look at the convergence of two different sums: that of the reciprocals of the primes with a prime number of digits, and that of the reciprocals of the natural numbers which do not contain your phone number amongst their digits.…

By Dylan Nelson| May 22nd, 2016|Level: intermediate, Uncategorized|2 Comments

Nowhere Differentiable Functions

By: Jan Wuzyk

In this article I am going to show that nowhere differentiable functions do in fact exists and give a few examples, some of which are relatively modern. But first I’m going to try to answer a question that is, in my opinion, too rarely discussed in mathematics classes, ”Why do we care?”.

Why we care

To answer this question we have to look into the history of mathematics. In 1821 Augustin-Louis Cauchy published his seminal book, Cours d’Analyse, this is generally recognized as the first serious attempt to put calculus on a rigours footing[Com]^[1] , mainly through introducing rigorous definitions of limits,continuity and differentiability among others, and the definitions that go with them². This was also time the integral was defined as an area instead of simply as the antiderivative.

It should be noted that Cauchy by no means closed the issue of rigour in analysis but he provided a starting point.…

By Crypto Giants| May 17th, 2016|Uncategorized|0 Comments

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The Newton-Raphson Method

Courses, First year, MAM1000, Uncategorized, Undergraduate

The Newton-Raphson Method

How would you go about finding the value of $\sqrt{3}$ if you didn’t have a square root button on your calculator? Well, the most obvious thing might be to try some values, based on your knowledge of the square root function. You are being asked to find that x for which:

$x=\sqrt{3}$

or, in other words, that x, which, when squared gives 3. We have to be a little careful here because we know that there will actually be two numbers which satisfy this (one positive, one negative), and we are interested in the positive one only.

So, we try some values, but we don’t do it randomly, we can see that because $1^2=1$ and $2^2=4$ that whatever number squared gives 3 must be between 1 and 2. We can try something called a binary intersection. This just means taking the values which we know bound the right answer (ie. we know that 1<x<2), and trying the number in the middle.…

By Jonathan Shock| May 15th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|4 Comments

Welcome to Reproducing Kernel Hilbert Space

In a series of posts I hope to introduce Mathemafrica readers to some useful data analysis methods which rely on operations in a little back-water of Hilbert space, namely Reproducing Kernel Hilbert Space (or RKHS).

We’ll start with the “classic” example. Consider the data plotted in figure 1. Each data point has 3 “properties”: an $x_1$ coordinate, an $x_2$ coordinate and a colour (red or blue). Suppose we want to be able to separate all data points into two groups: red points and blue points. Furthermore, we want to be able to do this linearly, i.e. we want to be able to draw a line (or plane or hyperplane) such that all points on one side are blue, all points on the other are red. This is called linear classification.

Figure 1: A scatter of data with three properties: an x_1 coordinate, an x_2 coordinate and a colour.

Suppose for each data point we generate a representation of the data point $\phi(x)=[x_1, x_2, x_1x_2]$ .…

By Alex Antrobus| April 30th, 2016|English, Level: intermediate, Uncategorized|1 Comment

IMU Breakout Graduate Fellowship Program – Apply now!

The IMU (International Mathematical Union) has recently launched the novel IMU Breakout Graduate Fellowship Program.

Thanks to a generous donation by the winners of the Breakthrough Prizes in Mathematics – Ian Agol, Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terence Tao and Richard Taylor – IMU with the assistance of FIMU (Friends of the IMU) and TWAS (The World Academy of Sciences) has launched a fellowship program to support postgraduate studies in a developing country, leading to a PhD degree in the mathematical sciences. The IMU Breakout Graduate Fellowships will offer a limited number of grants for excellent students from developing countries. The program will be administered by CDC (Commission for Developing Countries), a commission of IMU.

Professional mathematicians are invited to nominate highly motivated and mathematically talented students from developing countries who plan to complete a doctoral degree in a developing country, including their own home country. Nominees must have a consistently good academic record from the high school level and must be seriously interested in pursuing a career of research and teaching in mathematics.

…

By Andreas Daniel Matt| April 27th, 2016|Uncategorized|0 Comments

Logical implications and the structure of if and only if statements

We had a homework assignment a couple of weeks back. It was looking at mathematics in a very different way from how many had seen it before, and it caused a lot of confusion. I would like to try and add some clarity to what we were doing. My thought was, rather than going through the questions themselves, I would like to add annotations to the proof itself. Let’s see how this works. The proof that you were given is in black, the annotations are in blue, and after I’ve been through the proof, I will expand on it in a simplified form.

Theorem: The function f is differentiable at x=a if and only if there is a constant m and a function E of x, defined for all $x \not = a$ , such that

$f(x)=f(a)+m(x-a)+E(x)(x-a)$ for all $x \not = a$ – (eq 1)

and,

$\lim\limits_{x\to a}E(x)=0$ .

(If both these conditions are satisfied, then $f'(a)=m$ .)

What we are doing here is giving another definition of differentiability (at a particular point, a).…

By Jonathan Shock| April 22nd, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

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